MQENet: A Mesh Quality Evaluation Neural Network Based on Dynamic Graph Attention

Mesh quality evaluation has been widely studied in various ways. Since it is difficult to define an evaluation function that takes the entire mesh as input, mesh quality evaluation typically utilizes element-based mesh quality metrics. Several mesh quality evaluation metrics have been proposed. Li et al. [16] summarized the mesh quality metrics for commonly used 2D and 3D meshes. For 2D mesh cells, their focus is mainly on two metrics: shape and size. 2D meshes are mainly include unstructured meshes and structured meshes. The metrics for 2D unstructured meshes primarily include element length, aspect ratio and skewness. The aspect ratio of a 2D unstructured mesh [17] is measured as:

where $L_{\mbox{max}(L_{0},L_{1},L_{2})}$ is the length of the longest side of the unstructured mesh. $L_{0},L_{1},L_{2}$ are the lengths of the sides of the unstructured mesh and $S$ is the area of the unstructured mesh.

Skewness [18] is a mesh angle metric that indicates how close a mesh cell is to an ideal cell:

where $Q_{\mbox{max}}$ and $Q_{\mbox{min}}$ are the maximum and minimum angles in the mesh element, respectively. $Q_{\mbox{ideal}}$ is the angle for an ideally shaped mesh element.

Although these mesh quality metrics are widely used in practice, their thresholds are often highly subjective, and it is difficult to evaluate the quality of a mesh cell from a comprehensive perspective.

In addition to the above mesh quality metrics, traditional machine learning methods are also used to evaluate mesh quality. Chetouani [19] proposed a 3D mesh quality metric based on feature fusion, which evaluates the 3D mesh quality by using the Support Vector Regression (SVR) model. Combined with the specified mesh quality metrics and geometric attributes, SVR is used to predict quality scores. Sprave et al. [20] extract low-level attributes through the neighborhood graph of the mesh. They consider the quality index of a mesh element by evaluating and summarizing the neighborhood quality index value. However, traditional machine learning methods generally have weak generalization, making it difficult to deal with data of different distributions after training.

In recent years, deep learning has been applied in various fields. To solve the above shortcomings, Chen et al. [11] proposed GridNet, a structured mesh quality evaluation based on CNNs. They also propose the structured mesh dataset NACA-Market. GridNet takes the mesh file as input and automatically evaluates the mesh quality. Chen et al. [12] also proposed an automatic hexahedral mesh evaluation framework, MVENet. It takes hexahedral mesh data as input, and studies the effect of mesh point distribution on numerical accuracy based on region segmentation and deep neural network. They employ a supervised learning process to fit the relationship between the hexahedral meshes and their quality. Wang et al. [13] proposed GMeshNet, a graph neural network to evaluate the quality of structured meshes, which converts the mesh quality evaluation task into a graph classification task.

To sum up, traditional mesh quality metrics have significant limitations. These metrics can only evaluate a single aspect of mesh quality and rely on the expertise of professionals. Traditional machine learning methods primarily rely on manual techniques to construct mesh features when evaluating mesh quality. Deep learning-based evaluation techniques struggle to handle dynamic topology in structured meshes, resulting in difficulties achieving satisfactory accuracy.

Recent research on analyzing graphs with machine learning has gained significant attention due to the expressive power of graphs [21]. GNNs are deep learning based methods that operate on the graph domain.

Graph Convolutional Networks (GCNs) are the core part of GNNs. GCNs exploit the graph structure and aggregate node information from neighborhoods in convolutional manners [22]. Graph Attention Network (GAT) is one of the most popular GCN architecture [23], which is considered to be a very advanced architecture in graph representation learning. Zheng et al. [24] used GAT to obtain different levels of representations to efficiently solve natural problems. Leeson et al. [25] proposed GRAVES, an algorithm selection technique in software based on GAT, which enables neural networks to make more accurate predictions by using several attention mechanisms. Cirstea et al. [26] proposed Graph-attention Recurrent neural Networks (GRN) to achieve accurate forecasting of time series.

Graph pooling is an essential component of GNNs. In order to obtain effective and reasonable graph representations, many scholars have proposed various types of graph pooling designs. Ying et al. [27] proposed a differentiable graph pooling module called DiffPool, which can generates hierarchical representations of graphs. It can be seamlessly combined with different GNNs’ architecture in an end-to-end manner. Ranjan et al. [28] proposed adaptive structure aware pooling, a sparse and differentiable pooling method that addresses the limitations of previous graph pooling architectures. Ma et al. [29] proposed a path integral-based graph neural network (PAN) for graph classification and regression tasks.

Taken together, the research on GNNs has been increasing, and it has been applied in some scenarios. Since meshes can be easily represented as graph data, it is most suitable to use GNNs to process meshes. However, to the best of our knowledge, there has been very little research on GNNs for mesh quality evaluation tasks. The potential application value of GNNs need to be further explored. Therefore, this paper proposes a structured mesh quality evaluation network based on graph attention. The proposed method improves the performance on mesh datasets compared with other approaches.

To extract more useful high-level features from mesh elements in structured meshes, we design a neural network based on dynamic graph attention. The architecture of MQENet is shown in Figure 1.

Our proposed MQENet employs a hierarchical structure, including graph convolutional layers, graph pooling layers, graph readout operations and a multi-layer perceptron. MQENet utilizes two efficient mesh preprocessing algorithms, the point-based graph with proximity distance and the element‑based graph with sparse operation, to convert meshes into graphs. GATv2 [30] and SAGPool [31] are selected as the graph convolution layer and graph pooling layer, respectively. The graph convolution layer learns the weight of neighbor nodes through a dynamic attention mechanism, which can realize the weighted aggregation of neighbor nodes. The graph pooling layer computes the projection score via a learnable vector. These scores are used to select the top-ranked nodes, thereby retaining the most valuable mesh elements. Finally, a Multi-Layer Perceptron (MLP) is used to classify the mesh quality.

GNNs are deep learning models used for processing graph data. This paper takes structured mesh data as input, so how to convert mesh data into graph data is a challenging task. Currently, two algorithms have been proposed [13], but they both have some problems. The point-based graph representations use only mesh nodes as graph nodes and adjacent edges as graph edges. This method does not consider the relationship between non-adjacent meshes. On the other hand, the element-based graph representation considers the diagonal elements as 1, which increases the complexity and time of mesh data conversion.

This paper proposes two efficient mesh preprocessing algorithms based on node and element representation. Inspired by Pfaff et al. [32], the point-based graph with proximity distance for structured mesh is designed. It introduces the concept of proximity distance on the basis of the previous point-based graph. For mesh data, a mesh not only have an adjacency relationship with the directly connected mesh, but also have adjacency relationships with other meshes that have relationships but are not connected. Proximity distance edges join other mesh nodes in the graph that have spatial proximity to the current mesh node by given a fixed distance, which enhances features in mesh data. By modifying the diagonal elements, the element-based graph with sparse operation is designed, which improve the efficiency of mesh data conversion.

Different from the tasks at the node level, the graph classification task requires global information of the graph data. This information includes both the structural information of the graph and the attribute information of each node.

In this paper, GATv2, based on the dynamic graph attention, is selected as the graph convolution layer in MQENet. The difference from GAT is that the dynamic graph attention assigns different attention scores for different nodes, while the scores obtained by the static attention mechanism are exactly the same. This shortcoming makes it difficult for GAT to distinguish the quality of different structured meshes and even hinders GAT from fitting the training data.

Some GNNs’ architecture regard the weights of all neighbors as the same. To solve this problem, GAT uses the score function $\alpha:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ to score node pairs $(h_{i},h_{j})$:

where $b\in\mathbb{R}^{2d^{\prime}},W\in\mathbb{R}^{d^{\prime}\times d}$ are learnable parameters, $\|$ denotes vector concatenation and LeakyReLU is a type of activation function based on a ReLU.

We can obtain a more expressive dynamic attention by modifying the order of operations in the GAT. The problem with the GAT is that $W$ and $b$ in its scoring function are used for calculations in sequence, and they are likely to collapse into a linear layer. To solve this problem, we can move $b^{T}$ out of the non-linear result before performing the operation:

In this way, GATv2 has powerful feature extraction capabilities for dynamic topology in structured meshes. For any two structured mesh elements or nodes, we can get the corresponding dynamic attention score $\alpha$, as shown in Figure 1. Additionally, LayerNorm [33] and LeakyReLU are performed. The feature matrix $X^{\prime}_{h}\in\mathbb{R}^{N\times n}$ input to the next pooling layer is:

where $X_{h}\in\mathbb{R}^{N\times m}$ and $A_{h}\in\mathbb{R}^{N\times N}$ are the feature matrix and the adjacency matrix from structured meshes, respectively. $N$ is the number of the graph nodes, $m$ is the number of input features and $n$ is the number of output features.

Since GNNs with a number of network layers (more than two layers) are very prone to over-smoothing problems, we added a residual connection mechanism [34] in the graph convolutional layer to prevent this problem:

Most graph pooling techniques need to calculate the distribution matrix of nodes in the process of clustering. However, the space-time complexity of the distribution matrix calculation algorithm is proportional to the nodes and edges of the graph. Mesh data may contain a large number of nodes and elements, which makes it difficult for most graph pooling techniques to be applied to mesh quality evaluation tasks. Thus, we adopt SAGPool to pool the graph data.

As shown in Figure 1, SAGPool adaptively learns the importance of nodes from the graph through graph convolution, and then uses the TopK [35] to discard nodes. Specifically, aggregation operations are used to assign an importance score to each node.

where tanh represents the activation function, $D_{h}\in\mathbb{R}^{N\times N}$is the degree matrix of $A_{h}$ and $W_{att}\in\mathbb{R}^{n\times 1}$ represents the weight parameter.

The pooling operation can be performed according to the importance score and the topology of the graph. Based on the score calculated by Equation 13, only $\lceil kN\rceil$ nodes are reserved.

where $k\in(0,1]$ is the pooling ratio.

Repeatedly stacking SAGPool can perform graph pooling, and finally reduce the dimensions of each graph to the same dimension. The reconstructed feature matrix and adjacency matrix are obtained by

To read out graph data from MQENet, we make graph data more complete by concatenating the global average readout operation and the global maximum readout operation. Due to the different node positions in the graph, it is difficult to obtain an accurate representation of each node. We choose JK-net [36] to obtain an accurate representation at different levels and realize the flexible use of different neighborhood ranges for each node to achieve better structure-aware representation. The final representation is the input to an MLP for classification. Additionally, BatchNorm [37] and LeakyReLU are adopted to stabilize the training process.

Currently, there are very few benchmark structured mesh datasets available for mesh evaluation tasks. Only two datasets, NACA-Market and AirfoilSet, exist. Among them, only NACA-Market is a public dataset. So we choose the NACA-Market dataset to evaluate MQENet. NACA-Market is a mesh quality evaluation dataset of NACA0012 airfoils with eight labels, totaling 10,240 meshes. Each subset of 1,024 meshes corresponds to airfoils of one size, with a total of ten sizes. All the meshes belong to eight different quality labels, with an average of 1,280 meshes per label. The eight labels consist of three structured mesh quality evaluation metrics, which are orthogonality, smoothness and distribution, as shown in Figure 4. In the experiments, we use ten NACA-Market subsets, which include 1,024 meshes and 8 label, to train MQENet separately and evaluate its performance.

After converting structured meshes into graphs, we set the division ratio of the training data, validation data and test data to 60%,20%,20%, and shuffle the data to ensure the distribution. We set the number of input features, output features, network layers and hidden layer units to 6, 8, 4 and 12, respectively. We select AMS-Grad [38] with an initial learning rate of 1e-2 as the optimizer, and dynamically decrease the learning rate according to the training situation. Furthermore, we use negative log likelihood loss as the loss function and add L2 regularization with 1e-4 weight decay. The batch size is set to 32. The pooling ratio for all pooling layers are set to 0.3. The experiments are carried out on an NVIDIA Tesla A100-40G.

We use the gradient clipping technique [39] to solve the gradient explosion problem that often occurs in neural networks. At the same time, the early stop method is adopted in the training process to find out the problems in the training process in time.

First, we evaluate the capabilities of MQENet on the test set of the NACA-Market dataset. The quality identification results of structured meshes with different characteristics are shown in Table 1. Among them, W represents the mesh without defects, N-O represents the mesh with poor orthogonality, N-S represents the mesh with poor smoothness, N-D represents the mesh with poor distribution, N-OS represents the mesh with poor orthogonality and smoothness and N-OSD represents the mesh with poor orthogonality, smoothness and distribution. It can be seen that MQENet achieve very good results. The accuracy is all above 81% on eight labels of structured meshes, and the highest can reach 83.31%. This illustrates the effectiveness of MQENet for structured mesh datasets.

Then, we can also see that when the mesh has multiple defects, MQENet has the potential to identify the result as a single defect. For example, in the N-OS results, MQENet identified 11.74% of the meshes as N-O. There are similar problems in other labels. We believe that this is a problem of labeling in the dataset. Previous studies have directly divided the NACA-Market dataset into eight categories. In doing so, the eight categories are set as completely independent labels, and the relationship between labels are not considered, which leads to the above problems.

Finally, to prove the superiority of our method, we choose to perform comparative experiments with GMeshNet. We use two metrics, recall and accuracy, to measure the precision of the neural network. As shown in Table 2, we can see that the accuracy of GMeshNet trained on NACA-Market datasets is not as good as that of MQENet. In terms of orthogonality and smoothness, the recall can reach 82.96% and 81.64%, which is higher than GMeshNet. In terms of test accuracy, it reaches 82.67%, which is also better than GMeshNet. More importantly, MQENet can process mesh data faster than GMeshNet.

In summary, compared to other methods, our proposed MQENet has better accuracy and recall. This demonstrates that MQENet can perform well on structured mesh quality evaluation tasks. The network structure we designed can well adapt to the dynamic topology in different structured meshes. By filtering important nodes, hidden features in structured meshes can be more accurately captured. So that MQENet can better classify structured meshes with different quality labels in the feature space.

In this section, to illustrate the effectiveness of MQENet, this paper performs ablation studies from three aspects: mesh preprocessing algorithms, hyper-parameters and network structure.